ILL 1986 Problem 59

Let $ABCD$ be a convex quadrilateral whose vertices do not lie on a circle. Let $A'B'C'D'$ be a quadrangle such that $A',B', C',D'$ are the centers of the circumcircles of triangles $BCD,ACD,ABD$, and $ABC$. We write $T (ABCD) = A'B'C'D'$. Let us define $A''B''C''D'' = T (A'B'C'D') = T (T (ABCD)).$

(a) Prove that $ABCD$ and $A''B''C''D''$ are similar.

(b) The ratio of similitude depends on the size of the angles of $ABCD$. Determine this ratio.